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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.4.16
Chapter 8, Problem 8.4.16

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)² dx

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Step 1: Recognize that the integral involves a term of the form (a² + x²). This suggests using the trigonometric substitution x = a * tan(θ), where a = 5 in this case. Substitute x = 5 * tan(θ), which implies dx = 5 * sec²(θ) dθ.
Step 2: Substitute x = 5 * tan(θ) into the integral. Replace x² with (5 * tan(θ))² = 25 * tan²(θ), and replace dx with 5 * sec²(θ) dθ. The denominator (25 + x²)² becomes (25 + 25 * tan²(θ))² = (25 * sec²(θ))².
Step 3: Simplify the integral using the trigonometric identities. The integral becomes ∫ (25 * tan²(θ)) / (625 * sec⁴(θ)) * (5 * sec²(θ)) dθ. Simplify the expression by canceling terms and reducing powers of sec(θ).
Step 4: After simplification, the integral reduces to ∫ (tan²(θ) / sec²(θ)) dθ. Use the identity tan²(θ) = sec²(θ) - 1 to rewrite the integral as ∫ (sec²(θ) - 1) / sec²(θ) dθ = ∫ (1 - 1/sec²(θ)) dθ.
Step 5: Split the integral into two parts: ∫ 1 dθ - ∫ 1/sec²(θ) dθ. Evaluate these integrals separately. The first integral ∫ 1 dθ is θ, and the second integral ∫ 1/sec²(θ) dθ is tan(θ). Finally, back-substitute θ using the original substitution x = 5 * tan(θ) to express the result in terms of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots or quadratic expressions. By substituting a variable with a trigonometric function, such as x = a tan(θ) or x = a sin(θ), the integral can often be transformed into a more manageable form. This method leverages the identities of trigonometric functions to facilitate integration.
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Integral of Rational Functions

Integrating rational functions involves finding the antiderivative of a fraction where both the numerator and denominator are polynomials. Techniques such as polynomial long division, partial fraction decomposition, or trigonometric substitution can be employed to simplify the integral. Understanding how to manipulate these functions is crucial for solving integrals like ∫ x²/(25 + x²)² dx.
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Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is often used in trigonometric substitution to relate different trigonometric functions and simplify expressions. When substituting variables, recognizing how to apply this identity can help in transforming the integral into a solvable form.
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Related Practice
Textbook Question

7–84. Evaluate the following integrals.

11. ∫ from 0 to π/4 (sec x – cos x)² dx

Textbook Question

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.

44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.

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Textbook Question

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.

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Textbook Question

23-64. Integration Evaluate the following integrals.

26. ∫₀¹ [1 / (t² - 9)] dt

Textbook Question

59. Perpetual Annuity

Imagine that today you deposit $B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of $I per year from the account forever. The amount of money that must be deposited is:

B = I × ∫(from 0 to ∞) e^(-rt) dt

where r = p/100.

Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of \$5000 per year. How much must you deposit today?

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Textbook Question

7–84. Evaluate the following integrals.

35. ∫ from 0 to π/4 [(tan²θ + tanθ + 1) sec²θ] dθ